This project consists of two intertwined components: (a) development of new statistical methods that address recurrent problems in the analysis of functional neuroimaging data and (b) neuroscientific studies of visual remapping in human cortex, which will frame the need for and guide the development of our new methods. The new statistical methods in this proposal apply directly to diverse applications beyond functional neuroimaging, from galaxy clustering to DNA microarrays. The three neurosciences questions in this proposal address fundamental issues regarding neural mechanisms of remapping in humans. Visual remapping is the process that coordinates the visual and eye-movement systems in order to maintain stable perception of the world when the eyes move. This project will achieve the following specific aims, each tied to a specific scientific question. Aim 1. To develop new multiple testing methods for false discovery control. Question 1. Does remapping occur outside parietal cortex? Extending recent work on controlling the False Discovery Rate this project will develop procedures that bound the unobserved proportion (or number) of false discoveries with specified confidence. The method will be applied to investigate Question 1. Aim 2. To develop tools for finding nonlinearly optimal fMRI designs. Question 2. What is the time course of remapping? This project will implement design tools that optimize targeted inferences -- linear and nonlinear. The tools will be applied to design experiments that address Question 2. Aim 3. To develop nonparametric confidence sets for structured function estimation. Question 3. How do the shapes of the visual and remapped responses differ? This project will build on recent work in nonparametric regression to construct confidence sets for an unknown function under shape constraints and with dependent noise. The method will be applied to Question 3.